The development of the model and arithmetic for the fully distributed fiber optic sensor based on Raman optical-fiber frequency-domain reflectometry (ROFDR)

Sensors and Actuators A 101 (2002) 132±136

The development of the model and arithmetic for the fully distributed ®ber optic sensor based on Raman optical-®ber frequency-domain re¯ectometry (ROFDR) Junping Geng*, Jiadong Xu, Yan Li, Gao Wei, Chenjiang Guo Department of Electronic Engineering, Northwestern Polytechnic University, 710072 Xi'an, Shaan Xi, China Received 26 January 2002; received in revised form 10 May 2002; accepted 27 June 2002

Abstract The theoretical model of Raman optical-®ber frequency-domain re¯ectometry (ROFDR) which is brought forward in these years is analyzed, and the defect of the model is found. The theoretical model is consummated by regarding the Bose±Einstein factor as the function of temperature. Because the temperature changes slowly, the relation between the frequency and time is studied, and the model between the u domain and z domain is given. The emulational result is presented. By the corrected method, the corrected calculation temperature is almost accordant with the real temperature, and the error is less than that is reported. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Raman optical-®ber frequency-domain re¯ectometry (ROFDR); Distributed temperature ®ber optic sensor

1. Introduction Optical-®bers recently have become important elements in communication systems and sensor systems due to their high ef®ciency of light transport and their large bandwidth. These properties have made optical-®ber sensor increasingly more important instead of the traditional sensors. Especially, people pay more attention to the distributed optical-®ber sensor for its high resolution and less error [1]. In those years, the distributed optical-®ber sensor on the base of optical time-domain re¯ectometry (OTDR) and quasi-distributed method was the center of research. Recently, with Raman scattering being lucubrated, some researchers turned their interest to Raman distributed sensors which are based on the Raman optical timedomain re¯ectometry (ROTDR) [2], a short laser pulse is sent along the ®ber and the backscattered Raman light is detected with high temperature solution, the backscattered light including the information of light loss and temperature along the ®ber. Another new distributed temperature optical-®ber sensor is Raman optical frequency-domain re¯ectometry (ROFDR)

* Corresponding author. Tel.: ‡86-29-8491-414; fax: ‡86-29-8491-000. E-mail address: [email protected] (J. Geng).

[3] sensor, which is based on the Raman optical frequencydomain re¯ectometry [1]. It is invented by Farahani and Gogolla in 1998. But there is some limitation and defect in their model, and only the unique temperature is analyzed in their paper. In this paper, we analyzed the Raman scattering and OFDR, and get an consummated ROFDR model. We can process any temperature distribution by this model. 2. Raman light scattering In the frequency-domain, photons of Raman scattering include Stokes scattering photon and anti-Stokes scattering photon [4,5]. Their frequency can be expressed as follows. Stokes scattering frequency: fS ˆ f0

Df

(1)

Anti-Stokes scattering frequency: fAS ˆ f0 ‡ Df

(2)

Here, f0 is the frequency of incident light, Df is the frequency shift. For the fused silica, Df ˆ 1:32  1012 Hz [6]. During the Raman scattering, the photon jumped from the beginning steady-state to another steady-state, the energy difference DE ˆ h Df .

0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 2 ) 0 0 2 0 8 - X

J. Geng et al. / Sensors and Actuators A 101 (2002) 132±136

3. Analysis and consummation of Raman optical frequency-domain reflectometry for distributed temperature

133

According to the Bose±Einstein probability distribution of photons, the Bose±Einstein factor of the Stokes line is [1,8]: 1 (5) rS ˆ 1 exp… DE=kT†

3.1. Principle picture Fig. 1 is the basic scheme of Raman optical frequencydomain re¯ectometry (ROFDR) for distributed temperature. The probe laser power of frequency f0 is coupled into the ®ber at the position z ˆ 0. The light of this laser is sinusoidally intensity-modulated by an electrooptic modulator (EOM). Here, equidistant modulation frequency fm is used. The modulator is driven by a signal generator (SG). The powers of Stokes and anti-Stokes lines are detected by avalanche photodiodes (APD). One percent of the input probe power is detected by a pin photodiode (PPD). The output signals of the photodiodes, which are proportional to the modulated optical powers, are fed to a signalprocessing system. The distribution temperature is gotten in the end.

where k is the Boltzmann's constant, and T the absolute temperature of the ®ber. If T is the distribution temperature, it is the function of time t and position z. Temperature changed little in short time (relative to the measurement time), so that rS and T are just the function of position z. This means

3.2. Consummated Raman scattering model

rAS ˆ

For multimode step-index ®bers, the effective crosssection of the ®ber core is A, and the molecule density of the ®ber core is N0. There is N0A dz molecules in the volume of A dz, and the total Raman scattering power is

dP0AS;step ˆ rAS GAS;step P0 dz  dseff p ; oAS0 GAS;step ˆ pb2c N0 dO 2

dP0S;step ˆ rS GS;step P0 dz

rAS …z† ˆ

Here, Raman Stokes capture coef®cient [1] GS;step is  dseff p ; oS0 GS;step ˆ pb2c N0 dO 2

(3)

rS …z† ˆ

1

1 exp… DE=kT…z††

(6)

Eq. (3) can be written as dP0S;step ˆ rS …z†GS;step P0 dz

(3a)

In (3a), Stokes Raman scattering power is related with the distribution temperature. The anti-Stokes line is given in a quite similar way as [9]: dP0AS;step ˆ rAS GAS;step P0 dz 1

(7)

exp… DE=kT† exp… DE=kT†

(7a) (7b) (8)

exp… DE=kT…z†† (9) 1 exp… DE=kT…z†† In (7b), anti-Stokes Raman scattering power is related with the distribution temperature too.

(4)

where (dseff/dO) (p/2, oS0) is differential Raman crosssections [1,5], which has no relation with the temperature. The bc [7] is the maximal angle between the wave vector of the scattered light and the ®ber axis.

4. Optical-fiber Raman scattering with modulated probe light We assume that the optical-®ber axis direction is z, the input laser power at z ˆ 0 is sinusoidally intensity-modulated in form of ^ 0 ‰1 ‡ cos…om t†Š P0 …t; z ˆ 0† ˆ P

(10)

The amplitude and phase in (10) are very weakly dependent on the angular modulation frequency om. Consequently, this dependence is neglected. While the modulated probe laser power is traveling through the ®ber, it experiences loss due to ®ber attenuation and a phase shift. Fiber dispersion is another important factor, it implies a low pass LP;1 (forward direction) of the probe light. Hence, the probe power as a function of the ®ber location z can be expressed by Fig. 1. Basic scheme of Raman optical frequency-domain reflectometry (ROFDR) for distributed temperature measurement.

^ 0 exp‰ aP …l0 †zŠ‰1 ‡ LP;1 cos…om t P0 …t; z† ˆ P

km z†Š (11)

134

J. Geng et al. / Sensors and Actuators A 101 (2002) 132±136

If the Raman Stokes scattering happen in a little sector dz at location z, we measure the Raman Stokes back scattering light power at the input end of ®ber; the power can be expressed by ^ 0 rS …z†GS exp… ‰aP …l0 † ‡ aP …lS †Šz† dPS;step …0; z† ˆ P  ‰1 ‡ RefLP;1 LP;2 exp…jom t 2jkm z†gŠ dz

(12)

where LP;2 implies a low pass (backward direction) of the probe light. Assuming the length of the multimode optical-®ber is L, the modulation frequency is less than 100 MHz [10], then measured Raman Stokes back scattering power of the whole ®ber is Z L ^ 0 GS rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz† dz PS;step …0; om † ˆ P 0 Z L ^ 0 GS rS …z† Refexp…jom t 2jkm z ‡P 0

‰aP …l0 † ‡ aP …lS †Šz†g dz

(13)

If we divide the optical-®ber into N sectors in z direction, the length of every sector is DL, z coordinate of every node is z…0† ˆ 0; . . . ; z…i† ˆ i DL; . . . ; z…N† ˆ L. Because the temperature changes slow in time, we can assume that the temperature T(i) of the ®ber sector …z…i†; z…i ‡ 1†† is constant if we divide the ®ber enough densely. For the input light wavelength l0, lS of Raman Stokes scattering wavelength is a certain value, we can write AS ˆ aP …l0 † ‡ aP …lS †. For every modulated frequency om, Eq. (13) can be expressed as Z L ^ 0 GS rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz† dz PS …0; om † ˆ P 0 Z L ^ 0 GS rS …z† exp… ‰aP …l0 † ‡ cos…om t† P 0 Z L ^ 0 GS rS …z† ‡ aP …lS †Šz†cos…2km z†dz‡sin…om t†P 0

 exp… ‰aP …l0 † ‡ aP …lS †Šz† sin…2km z† dz Z L ^ 0 GS rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz† dz ˆP 0 p ^ 0 GS A2 ‡ B2 cos…om t g† ‡P It means g†

(14a)

Here, Z L Aˆ rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz† cos…2km z† dz 0

0

L

rS …z† exp… ‰aP …l0 †‡aP …lS †Šz† sin…2km z† dz

  B g ˆ arcsin p A2 ‡ B2 Z L S ˆ P ^ 0 GS P rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz† dz 0

^S ˆ P ^ 0 GS P

p A 2 ‡ B2

(14c) (14d) (14e) (14f)

So, the response function of system in frequency-domain can be written as ^S P HS …jom † ˆ exp… jg† ˆ GS …A jB† ^0 P Z L ˆ GS rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz† 0 Z L GS rS …z†  ‰cos…2km z† j sin…2km z†Š dz ˆ 0

 exp… ‰aP …l0 † ‡ aP …lS †Šz† exp… j…2km z† dz (15)

5. Signal processing and analyzing

S ‡ P ^ S cos…om t PS …0; om † ˆ P

Z Bˆ

In (15), if we expand the integral range of z to ( 1, ‡1), and assume u ˆ 2km (2nco/c)om, (15) can be written as Z ‡1 fGS rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz†g HS …ju† ˆ 1

 exp… juz† dz

(16)

Here, (16) is just the fast Fourier transform (FFT) from z to u. By the inverse fast Fourier transform (IFFT), hS(z) can be given as hS …z† ˆ RefIFFT…HS …ju††g ˆ GS rS …z† exp… ‰aP …l0 † ‡ aP …lS †Šz†

From (l4) to (17), the relation between response function hS(z) in z domain and HS(ju) in u domain is given. This method is similar to anti-Stokes photon too, the response function hAS(z) in z domain and HAS(ju) in u domain can be written as Z ‡1 HAS …ju† ˆ fGAS rAS …z† exp… ‰aP …l0 † ‡ aP …lAS †Šz†g 1

 exp… juz† dz

(18)

hAS …z† ˆ RefIFFT…HAS …ju††g ˆ GAS rAS …z† exp… ‰aP …l0 † ‡ aP …lAS †Šz†

(19)

Then dividing (17) by (19): hS …z† GS rS …z† ˆ exp… ‰aP …lS † hAS …z† GAS rAS …z†

aP …lAS †Šz†

Taking (6) and (9) into (20):   hS …z† GS DE ˆ exp exp… ‰aP …lS † hAS …z† GAS kT…z† (14b)

(17)

(20)

aP …lAS †Šz† (20a)

J. Geng et al. / Sensors and Actuators A 101 (2002) 132±136

135

Then, (20a) can be solved: T…z† ˆ

DE k lnf…hS …z†=hAS …z††…GAS =GS †exp…‰aP …lS † aP …lAS †Šz†g

…21†

6. Calculation result and analyzing We assume the ®ber length is 1000 m, modulation frequency step is 10 kHz, input wave length is 1320 nm. When the real temperature of the whole ®ber is 273, 293, 313, 333, 353, 373 or 393 K, we get the calculation result by the earlier arithmetic in Figs. 2 and 3. In Fig. 2, the calculation temperature is lower (46±46.2 K) than the real temperature (273 K). The calculation result is almost line, and the maximal wave-error is 0.07919 K in the whole ®ber. These mean that calculation result is similar with the real temperature, but lower because of system error and parameter error. In Fig. 3, these real temperature have same interval 20 K, all the calculation result are lower than the real, and have

Fig. 4. Comparison of real T ˆ 273 ‡ 0:2z with calculation result.

little wave-error. But they all are almost line, and reasonable in the magnitude order. In Fig. 4, the real temperature is T ˆ 273 ‡ 0:2z, the calculation result is less than the real, but its distribution trend and curve is similar with the real. In Fig. 5, maximal wave-error Te of the calculation temperature increases nearly linearly with the real temperature, but the slope is very little (less than 5:239167  10 4 ). From 273 to 393 K, the maximal wave-error is from 0.07919 to 0.14206 K. In Fig. 6, the average deviation Td (between the calculation temperature Tc and the real temperature T) increases with T. By second order polynomial ®tting: Td ˆ

10:30784 ‡ 0:10204T ‡ …3:83796E

4†T 2

(22)

Correcting the calculation temperature Tc, T ˆ Tc ‡ Td , so …3:83796E Tˆ Fig. 2. The calculation temperature of the real temperature which is 273 K.

Fig. 3. Variable calculation result for real temperature T ˆ 273, 293, 313, 333, 353, 373, 393 K, and the real temperature T ˆ 273 K.

0:89796

4†T 2

0:89796T ‡ …Tc

10:30784† ˆ 0

p 0:897962 4…3:83796E 4† …Tc 10:30784† …7:67592E 4†

(23)

Fig. 5. Maximal wave error of calculation temperature Te  Re(T).

136

J. Geng et al. / Sensors and Actuators A 101 (2002) 132±136

Only if the calculation precision is high enough, we can detect and analyze any temperature distribution along the ®ber on base of improved ROFDR model and corresponding appropriate signal-processing method.

References

Fig. 6. The average deviation Td  Re(T).

Fig. 7. Comparison of the corrected calculation temperature with the real temperature (T ˆ 393 K).

[1] M.A. Farahani, T. Gogolla, Spontaneous Raman scattering in opticalfibers with modulated temperature Raman remote sensing, J. Lightwave Technol. 17 (1999) 1379±1391. [2] J.P. Geng, J.D. Xu, C.J. Guo, G. Wei, The development and trend of fully distributed fiber optic sensor for distributed temperature measurement, J. Transducer Technol. 20 (2001) 4±8 (in Chinese). [3] H. Ghafoori-Shiraz, T. Okashi, Fault location in optical-fiber using optical frequency-domain reflectometry, J. Lightwave Technol. LT-4 (1986) 316±322. [4] Z.X. Zhang, X.D. Yu, N. Guo, X.B. Wu, The optimum design of distributed optical-fiber Raman photon sensor (DOFRPS) system, J. Optoelectron. Laser 10 (1999) 110±112 (in Chinese). [5] G.X. Cheng, Raman and Brillouin Scattering: Principle and Application, Science Press in China, Beijing, 2001, p. 74 (in Chinese). [6] F.L. Galeener, J.C. Mikkelsen, R.H. Geils, W.J. Mosby, The relative Raman cross-section of vitreous SiO2, GeO2, B2O3 and P2O5, Appl. Phys. Lett. 32 (1978) 34±36. [7] A.W. Snyder, J.D. Love, Optical Waveguide Theory, J.W. Arrowsmith Ltd., Bristol, UK, 1983, p. 923 (in Chinese). [8] Z.X. Zhang, T.F. Liu, B.X. Zhang, Laser Raman spectrum of opticalfiber and the measurement of temperature field in space, Proc. SPIE 2321 (1994) 186±190. [9] D.A. Long, Raman Spectroscopy, McGraw-Hill, New York, 1977, p. 106 (in Chinese). [10] H. Ghafoori-Shiraz, T. Okashi, Optical-fiber diagnosis using opticalfrequency-domain reflectometry, Opt. Lett. 10 (1985) 160±162.

Biographies In Fig. 7, the real temperature 393 K and the corrected temperature are given in the whole ®ber. The maximal error is only 0.2 K, which is less than the reported result by others [1]. The correct calculation result is consistent with the practical temperature distribution, which means that the model and signal-processing methods in this paper are applicable. 7. Conclusions In this paper, an improved ROFDR theoretical model is presented based on the early ROFDR [1] model and the corresponding appropriate signal-processing method is obtained. It is very easy to measure because the measure time delay t, and higher resolution can be achieved by the improved ROFDR model than by ROTDR. The emulational result by the improved ROFDR model is consistent with the practical temperature distribution, and the error is less than the result of others.

Junping Geng was born in Bao Ji county, Shaan Xi province, China, in 1972. Since 1999, he has been working toward the PhD degree at the Department of Electronic Engineering, Northwestern Polytechnic University, China. His research interests are distributed optic fiber sensors and smart organ. Jiadong Xu was born in An Hui province, China. He is a professor and PhD advisor of the Department of Electronic Engineering, Northwestern Polytechnic University, China. His research interests are optics and microwave technology. Yan Li was born in He Nan province, China. Since 2000, he has been working toward the PhD degree at the Department of Electronic Engineering, Northwestern Polytechnic University, China. His research interests are digital signal process and transmission. Gao Wei was born in Shaan Xi province, China. Since 2001, he has been working toward the PhD degree at the Department of Electronic Engineering, Northwestern Polytechnic University, China. His research interests are microwave signal process and transmission. Chenjiang Guo was born in Shaan Xi province, China. Since 2001, he has been working toward the PhD degree at the Department of Electronic Engineering, Northwestern Polytechnic University, China. His research interests are microwave signal process and transmission.